I understand that most points will be close to surface due to volume concentration. Also I also understand the concentration of volume near the equator, relative to any specific point (North pole).
...

I’m a little confused, for a chain to be acyclic, all Betti numbers must be zero. For a Betti number $\beta$
$\beta_i=\dim(Z_i)-\dim(B_i)$ where $Z_i=\ker(\partial_i)$ and $B_i$=im$(\partial_i{+}_1))$...

I need A little bit more clarification when computing the homology of a chain complex. So the problem is:
Compute the simplicial homology of the graph with vertices $$V=\left\{ 1, 2, 3, 4 \right\}$$ ...

Say we have an essential class (barcode $[a,\infty)$, meaning its represents a feature that never gets killed) represented in a persistence diagram on $\bar{\mathbb{R}}^2$ as x= $(a,\infty)$. Where $\...

Do the homology groups for each Betti number as produced by persistent homology completely capture all the topological characteristics of the space? If not, what topological features are not captured? ...

I am very very new to Topological Data Analysis (TDA) and I am trying to apply the method described in this paper. I understood everything but stuck on methodology section (3.2) where he described:
...

I'm reading Zomorodian and Carlsson's paper on computing persistent homology. The object of their algorithm is, given a filtered complex, to find a representation of the kth persistent homology group ...

I am just beginning to learn about topological data analysis and understand the basics. With respect to constructing a persistence diagram, I understand level sets etc. My question is regarding how ...

I am planning on doing a project on topological data analysis in the near future and intend to use Gunnar Carlsson's paper "Topology and Data" as my introduction to the field. I am familiar with point-...

I was just thinking recently about if there are any possible meaningful connections between tools such as persistent homology used for things like topological data analysis and tools used in spectral ...

The reduction algorithm (pg. 5 of http://geometry.stanford.edu/papers/zc-cph-05/zc-cph-05.pdf) enables us to compute homology for modules over a PID.
I am curious why the reduction algorithm cannot ...

Does anyone know if persistent homology with integer coefficients are being used anywhere?
From what I understand, Carlsson's persistent module theory (http://citeseerx.ist.psu.edu/viewdoc/download?...

Q1) In the paper ([Zomorodian, page 6 https://pdfs.semanticscholar.org/b7ae/132ef88bde28903ac8b14a29a76130f61ac2.pdf), it is stated without proof that $im\ \eta_k^{i,p}\cong H_k^{i,p}$. May I know how ...

I am relatively new in the field of persistent homology and topological data analysis. I would like to use RIPSER, DIPHA or GUDHI to calculate barcodes which will give a persistence diagram. Here are ...

I am interested in seeking out a reference for learning Persistence Homology (more of the theoretical mathematical aspects rather than the applications/ computer science aspects).
Currently, the two ...

I am writing some code to generate a Vietoris-Rips complex given a set of data points. The VR complex is generated from 0-complexes, 1-complexes, 2-complex, etc., where $k+1$ represents the number of ...

Can anyone describe to me in Layman's terms what kind of use does Topology have through Persistent Homology in Data Analysis...that's can you give me some real life examples in which which this has ...

I am starting to do some Topological Data Analysis and am looking for a good plotting library to produce the diagrams of complexes, barcodes, and persistence diagrams. The most popular TDA libraries ...

I'm trying to understand the difference between Oracle's graph technology, which apparently has an inherent understanding of topology, and Ayasdi's Topological Data Analysis technology. Are these two ...

I often read how Topological Data Analysis (TDA) is useful especially for highly dimensional data. But, what about (apparently) low dimensional ones?
Example: consider measuring the resistance of a ...

I was wondering if anyone could help me out with finding a nice introductory introductory text for topological data analysis (I'm speaking as somebody who has two semesters of experience with topology,...